Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31996 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28243 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,14)(5,8)(6,21)(7,11)(9,17)(10,23)(13,28)(15,18)(16,27)(19,20)(22,26)(24,31)(25,30)(29,32);(2,8)(3,13)(4,11)(5,14)(6,20)(9,25)(10,21)(12,26)(15,32)(16,29)(17,22)(18,30)(23,31)(27,28);(1,3,14,17,7,18,5,16)(2,9,4,15,11,27,8,12)(6,22,31,13,23,29,20,30)(10,25,19,26,21,28,24,32);(1,4)(2,5)(3,15)(6,19)(7,8)(9,16)(10,20)(11,14)(12,18)(13,28)(17,27)(21,31)(22,26)(23,24)(25,30)(29,32)Polycyclic presentation:
2^9|c^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#510872, #72320498, … |
| maximal normal subgroup of | #72237280, C2×#510872, #72320498, … |
| maximal quotient of | C2×#510872, #72250548, #72250551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418068);